Positive dispersion: 2 n. λ 2 > 0. ω 2 > 0, Negative dispersion: ω < 0, 2 n

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Martha Fredriksen
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2 Positive dispersion: 2 n ω 2 > 0, 2 n λ 2 > 0 Negative dispersion: 2 n ω < 0, 2 n 2 λ < 0 2

3 φ(z,ω) = k ( n ω )z E( z,t)= 1 2π E ( z = 0,ω )e iωt iφ z,ω e ( ) dω

4 φ(z,ω) = k ( n ω )z φ( ω )= φ 0 + ω ω 0 ( ) dφω ( ) E( z,t)= 1 2π dω + ω0 E ( z = 0,ω )e iωt iφ z,ω e ( ) dω ( ) 2 d 2 φω ( ) 1 2! ω ω 0 dω ω0 ω 0

5 φ(z,ω) = k ( n ω )z φ( ω )= φ 0 + ω ω 0 ( ) dφω ( ) E( z,t)= 1 2π dω + ω0 E ( z = 0,ω )e iωt iφ z,ω e ( ) dω ( ) 2 d 2 φω ( ) 1 2! ω ω 0 dω ω0 ω 0

6 φ(z,ω) = k ( n ω )z φ( ω )= φ 0 + ω ω 0 ( ) dφω ( ) E( z,t)= 1 2π dω + ω0 E ( z = 0,ω )e iωt iφ z,ω e ( ) dω ( ) 2 d 2 φω ( ) 1 2! ω ω 0 dω ω0 ω 0

7 t p (0) t p (z) t t τ p ( z) τ p ( 0) = 1 + 4ln2 d 2 φ dω 2 τ 2 p ( 0) 2 d 2 φ dω >> τ 2 2 p( 0) τ p () z d 2 φ dω Δω 2 p

8 p τ τ d 4ln2 d 1 p p ϕ ω τ τ τ = +

9 n 2 1 = B k λ 2 Positive dispersion: 2 n ω 2 > 0, 2 n λ 2 > 0 k λ 2 C k up-chirped dω dt > 0 B 1 = , C 1 =(65 nm) 2, B 2 = , and C 2 =(132 nm) 2

10 τ p () z d 2 φ dω Δω 2 p

11 n= n() λ

$Refraction is$

12 Dispersion of material n( ) Snellius Law: Refraction is frequency dependent d.h. = ( )

13 t t

16 J. A. Valdmanis et al., Opt. Lett. 10, 131, 1985

17 Pump-Laser Ti:Saphir Kristall (2.3 mm, 0.25 wt.%) M 2 M 3 M 4 SESAM M 1 M 7 M 5 M 6 Quartzprismen Auskoppel- Spiegel

18 θ 1 ( λ 0 )= θ 2 ( λ 0 )= α 2 tanθ B = n 2 = n, and θ B + θ B = 90 o n 1 α = 180 o 2θ B = π 2θ B

19 n n = sin αδ ( min )+ δ min 2 ( ) sin αδ min 2

20 θ 1 ( λ 0 )= θ 2 ( λ 0 )= α 2 tanθ B = n 2 = n, and θ B + θ B = 90 o n 1 θ 2 ( λ)= arcsin( nsin θ 2 )= arcsin n( λ)sin π 2θ B arcsin sinθ B n( λ) α = 180 o 2θ B = π 2θ B

21 C θ 2 G H A β L P = 2 CDE opt = 2 AB = 2Lcosβ β θ 2 D E F s h B φ p = kp ( λ)= 2π λ P ( λ) d 2 φ P dω 2 = λ 3 2π c 2 d 2 P dλ 2 d 2 P dλ 2 = P β 2 n β λ 2 n + 2 β n n 2 λ 2 + β n n 2 d 2 P λ dβ 2 β = 0 d 2 P dλ = 2 θ 2 2 n n λ 2 L 8 dn dλ 2 L < 0

22 EFG = BH CDE = AB = L cosβ E F G = BH= BH C D E = AB= L cos β dφ P dω = d dω kp(ω ) ( )= 2L c cosβ 2ω L c sinβ dβ dω, dβ dω < 0

23 Fused Quartz 0 SF10 d 2 φ/dω 2 [fs 2 ] d 2 φ/dω 2 [fs 2 ] L [m] L [m]

24 d 2 φ dω >> τ 2 2 p( 0) Fused Quartz SF10 d 2 φ/dω 2 [fs 2 ] d 2 φ/dω 2 [fs 2 ] d 3 φ dω 3 >> τ p d 3 φ/dω 3 [fs 3 ] ( ) L [m] Fused Quartz d 3 φ/dω 3 [fs 3 ] L [m] 0.3 SF x L [m] L [m]

25 mm Fused Silica Prisms 35 cm apex separation 1000 adjustable negative GDD allows compensation of material dispersion strong higher-order contributions mm Wavelength (nm) 1000

26 λ < λ h λ > λ h Any wavelength < h will not be refracted by the second prism and will be lost during propagation through the prism pair. λ h

27 Ti:S-gain Prism + Ti:S-Crystal 0.8 GDD, fs x Standard Mirror (measured) Total Dispersion Spektrum x Chirped Mirror Wavelength, nm Goals: ---> High Reflectivity from 650 nm nm ---> Constant Total GDD from 650 nm nm

1.0 400 Ti:S-gain Prism + Ti:S-Crystal 0.

28 Ti:S-gain Prism + Ti:S-Crystal 0.8 GDD, fs x Standard Mirror (measured) Total Dispersion Spektrum x Chirped Mirror Wavelength, nm λ < λ h λ > λ h λ h

30 Λ ν x,m = m 1 Λ θ x,m m λ Λ t t t t

32 sinθ m = m λ Λ + sinθ i where m = 0, ± 1, ± 2,... x φ g ( x)= π m x Λ 2π

33 z L g x B θ i P' P Q b θ m A x sinθ m = m λ Λ + sinθ i where m = 0, ± 1, ± 2,... Beugungsgitter z.b. Blaze-Gitter φ = ω c L + φ g ( x), with x = L g tan( θ m ) φ g ( x)= π m 2π Λ x ( ) L = PABQ = PA+ b = b 1 + cos θ m + θ i = L g cosθ m ( ) 1+ cos θ m + θ i

34 sinθ m = m λ Λ + sinθ i L g d 2 φ dω = m2 λ 3 L g 2 2πc 2 Λ 1 m λ 2 Λ sinθ i 2 3/2 d 2 φ dω 2 L g angle of incidence angle of incidence 1200 lines/mm 300 lines/mm Λ 830 nm Λ 3.3 μm

35 L f stretcher something a Treacy grating pair cannot do! L f compressor like the Treacy grating pair

37 L f E out ( ω )= h ( ω ) E in ( ω )

39 A. M. Weiner, Rev. Sci. Instrum. 71, 1929 (2000) f = 300 mm knife-edge f = 300 mm SLM 640 pixels 300 l/mm grating 300 l/mm grating Possible bandwidth through Spatial Light Modulator: nm

48 R = 100% b AR-Coating R t R R GTI R GTI = 1 and R GTI = exp i2φ GTI ( ) d Glass substrate Air gap (non-absorbing spacer layer) r GTI exp( iφ GTI )= r 12 + r 23 e i2ϕ 1 + r 12 r 23 e = R t e i2ϕ i2ϕ 1 R t e i2ϕ t 0 t 0 = 2nd c ϕ = nkd = n ω c d ωt 0 2 dϕ dω t 0 2 tanφ GTI = Im r GTI Re r GTI ( ) ( ) = 1 R t ( )sin2ϕ 2 R t ( 1 + R t )cos2ϕ and d dω tanφ = ( 1 + tan2 φ) dφ dω ( ) R t sinωt 0 dt g dω = d 2 φ GTI = 2t R t dω 2 1+ R t 2 R t cosωt 0 ( ) 2

49 R = 100% b R t AR-Coating R R GTI dt g dω d 2 d Glass substrate Air gap (non-absorbing spacer layer) Bandwidth of dt g dω 1 d Example: d = 2.25 μm, R t = 4% Example: d = 80 μm, R t = 4%

50 t p (0) t p (z) t t L Substrate

Bragg-Mirror: SiO 2 - Substrate TiO / SiO 2 2 B - Layers 4 AIR 1.0 6 Intensity Reflexion 0.8 0.

51 Bragg-Mirror: SiO 2 - Substrate TiO / SiO 2 2 B - Layers 4 AIR Intensity Reflexion Phase Wavelength (μm) Wavelength (μm) 1.4

52 R λ/4 stack Substrate T Refr. index n H n S n I n L Physical distance r = ( n H n L ) ( n H + n L ) Δω ω B = 4 π arcsin( r) R 0 ( λ B )= 1 aqpm aqp m 1 2 p = n I n H q = n L n H a = n L n S

53 μ μ dt g dλ > 0 dt g dω < 0

54 λ Bragg Substrate Chirp Bragg wavelength wavelength-dependent penetration depth engineerable dispersion compensation of arbitrary material dispersion increased high-reflection bandwidth Group Delay (fs) Szipöcs et al., Opt. Lett. 19, 201 (1994) λ 4 λ 3 λ 2 GDD = 30 fs 2 (3.5 μm thick mirror stack) λ Wavelength (nm)

55 Substrate Gires-Tournois High Partial R = 5% Interferometer (GTI) Reflector Reflector Group Delay (fs) Wavelength (nm) GTI desired

56 Substrate Substrate AR Anti-reflection coating matches first-layer impedance to air Kärtner et al., Opt. Lett. 22, 831 (1997) Matuschek et al., IEEE J. Sel. Top. Quantum Electron. 4, 197 (1998)

57 k B,min = 2π 600 nm k B,max = 2π 900 nm

58 0 prism pair: tunability, better DCM designs (GDD= T g / ) 0 20 Double-Chirped Mirrors: trade off: bandwidth vs. GDD oscillations non-normal incidence: smoother average GDD (average of several DCMs) ) 2x 20 Wavelength (nm)

59 (a) Bragg Mirror: SiO 2 Substrate TiO / SiO 2 2 λ /4-Layers B Air (b) Simple-Chirped Mirror: Bragg Wavelength λ Chirped SiO 2 Substrate B λ 1 λ 2 Negative Dispersion: λ λ 2 > 1 GTI (c) Double-Chirped Mirror: Bragg Wavelength and Coupling Chirped SiO 2 Substrate AR Coating Air { Impedance Matching (n d < h h < n d ) l l

60 Δλ = 230 nm Wavelength (nm) With AR Without AR R AR < Wavelength (nm) 1000 cannot make arbitrarily low reflectivity and arbitrarily broad bandwidth at the same time J.A. Dobrowolski et al., Appl. Opt. 35, 644 (1996)

61 Substrate Perfect impedance matching with double-chirp and n substrate = n first layer Front-reflection from substrate is not interfering with wave from mirror stack (by geometry) AR coating required only for loss reduction

62 BASIC DCM: Target Design conventional DCM: Target Design Wavelength (nm) Wavelength (nm) BASIC conventional Wavelength (nm) 1400 R > 99.7% ( nm) T g variations = 0.3 fs rms GDD var. = 3.8 fs 2 rms 260 THz bandwidth

63 Error simulations Measured GDD Design Layer deposition error: 0.2 nm rms Wavelength (nm) 1000 future Error simulations Measured GDD Design Layer deposition error: 0.1 nm rms Wavelength (nm) 1000

64 N. Matuschek: Theory and design of double-chirped mirrors, Ph.D. Thesis, ETH Zurich (1999) Hartung-Gorre Verlag, ISBN

65 n o,k o = kn o, and ω,ω a,ω b [ ω 0 ± Δω ] k A,ω A n e, k e = kn e ω o + ω A = ω e, because ω A << ω ω o ω e k o + k A = k e k( n e n o )= k A = ω A υ A

67 + S ( Δt)= R E S * ()E t R ( t Δt)dt K. Naganuma, K. Mogi, H. Yamada, Opt. Lett. 15, (1990).

68 S ( ω )= E S ( ω ) E * R ( ω ) + S( Δt)= R E S ()E t * R ( t Δt)dt K. Naganuma, K. Mogi, H. Yamada, Opt. Lett. 15, (1990).

69 S ( ω )= E S ( ω ) E * R ( ω ) ϕω ( ) + S ( Δt)= R E S * ()E t R ( t Δt)dt K. Naganuma, K. Mogi, H. Yamada, Opt. Lett. 15, (1990). GDD = d 2 ϕω ( ) dω 2 ϕω ( )

70 S ( ω )= E S ( ω ) E * R ( ω ) + S ( Δt)= R E S * ()E t R ( t Δt)dt K. Naganuma, K. Mogi, H. Yamada, Opt. Lett. 15, (1990). GDD = d 2 ϕω ( ) dω 2 ϕω ( )

75 Phase Velocity υ p ω k n c n Group Velocity υ g dω dk n Group Delay T g Tg = z = dφ υ g dω, φ k n z c n nz c 1 1 dn λ dλ n dn λ 1 dλ n Dispersion 1. Order Dispersion 2. Order Dispersion 3. Order dφ dω d 2 φ dω 2 d 3 φ dω 3 nz c dn λ 1 dλ n λ 3 z d 2 n 2πc 2 dλ 2 λ 4 z 4π 2 c 3 3 d 2 n dλ + λ d 3 n 2 dλ 3

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